The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.

The phenomena of the real world, in domains as economics, biology, or environmental sciences, do not take place continuously, but at certain moments in time. Therefore, a discrete-time approach is required. By means of skew-evolution semiflows, we intend to construct a framework that deepens the analysis of discrete dynamical systems.

Playing an outstanding role in the study of stable and instable manifolds and in approaching several types of differential equations and difference equations, the exponential dichotomy for evolution equations is one of the domains of the stability theory with an impressive development. The dichotomy is a conditional stability, due to the fact that the asymptotic properties of the solutions of a given evolution equation depend on the location of the initial condition in a certain subspace of the phase space. Over the last decades, the classic techniques used to characterize asymptotic properties as stability and instability were generalized towards a natural generalization of the classic concept of dichotomy, the notion of trichotomy. The main idea in the study of trichotomy is to obtain, at any moment, a decomposition of the state space in three subspaces: a stable subspace, an instable one, and a third one called the central manifold. We intend to give several conditions in order to describe the behavior related to the third subspace.

A relevant step in the study of evolution equations is due to Henry, who, in [

A special interest is dedicated to the study of dynamic linear systems by means of associated difference equations, as emphasized by Chow and Leiva in [

In [

In [

The notion of trichotomy was introduced in 1976 by Sacker and Sell and studied for the case of linear differential equations in the finite dimensional setting in [

A stronger notion, but still in the finite dimensional case, was introduced by Elaydi and Hajek in [

In [

The notion of skew-evolution semiflow considered in this paper and introduced by us in [

The case of stability for skew-evolution semiflows is emphasized in [

The following sections outline the structure of this paper. In Section

Let us consider

The mapping

The approach of asymptotic properties in discrete time is of an obvious importance because the results obtained in this setting can easily be extended in continuous time.

Let

To every skew-evolution semiflow

This section aims to emphasize some asymptotic behaviors, as exponential growth and decay and exponential stability and instability, as a foundation for the main results. We give the definitions of these properties in continuous time and we underline the characterizations in discrete time, as results that play the role of equivalent definitions (see [

A skew-evolution semiflow

A skew-evolution semiflow

A skew-evolution semiflow

We have the following.

As a second step, for

Hence,

A skew-evolution semiflow

A skew-evolution semiflow

A skew-evolution semiflow with exponential decay

We have the following.

As a second step, if we consider

Hence,

A projector

Two projectors

projectors

for all

A skew-evolution semiflow

We denote by

Let

In what follows, let us denote

In discrete time, we will describe the property of exponential dichotomy as given in the next proposition.

A skew-evolution semiflow

for all

We have the following.

Statement

Hence,

A skew-evolution semiflow

there exist a constant

there exist a constant

for all

We have the following.

By Proposition

According to the hypothesis, if we consider

Three projectors

each projector

for all

A skew-evolution semiflow

We consider the evolution semiflow

We consider the projections

As in the case of dichotomy, let

In discrete time, the trichotomy of a skew-evolution semiflow can be described as in the next proposition.

A skew-evolution semiflow

for all

We have the following.

For

Let

For

By a similar reasoning, from

Hence, the skew-evolution semiflow

Some characterizations in discrete time for the exponential trichotomy for skew-evolution semiflows are given in what follows.

A skew-evolution semiflow

there exist a constant

there exist a constant

there exist a constant

there exist a constant

for all

We have the following.

Similarly, the other equivalences can also be proved.

In order to characterize the exponential trichotomy by means of four projectors, we give the next definition.

Four invariant projectors

A skew-evolution semiflow

for all

We have the following.

We will define the projectors

The statements of Proposition

The paper emphasizes a way to unify the analysis of continuous and discrete asymptotic properties for skew-evolution semiflows, such as the exponential dichotomy and trichotomy. Thus, we give characterizations for the asymptotic behaviors in discrete time, in order to gain necessary instruments in punctuating the properties of the solutions of difference equations.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author gratefully acknowledges helpful suggestions and support from Professor Emeritus Mihail Megan, the Head of the Research Seminar on